The standard color mode used by images before the advent of hdri is a bit limited. It only has colors in range of 0-1. But obviously colors and especially signal processing is not limited to a range of 0-1. Especially negative colors would be nice to have.
The overlay mode sidesteps this limitation. By setting 0.5 color as neutral, and anything below is subtracted (or that is what it emulates, not what it does) to the signal and adds the signal above 0.5. Since emboss is effect that wants to do adding a subtracting on both sides it is aimed for overlay and produces a grayish image with darker shadows and lighter highlights.
Now emboss is part of a family of a edge detection/highpass filters. Now it turns out that if you boost edges you in fact get apparent sharpening, much like unsharp mask. The image is not really sharper, the local contrast is higher. Which is much the same for less than 1 pixel sharpening, and since the human eyes that actually do similar kind of sharpening on the retina it will interpret the signal sharper.*
* But yes we can also really sharpen images since in the era of Hubble myopia, but that does increase the noise...